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Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods. (English) Zbl 1264.35260

This paper deals with the analysis and the implementation of numerical methods allowing to solve the null controllability problem for the following semi-linear one-dimensional heat equation: \[ y_t-(D(x)y_x)_x+f(y)=v1_\omega,\,(x,t)\in (0,1)\times (0,T), \]
\[ y(x,t)=0,\,(x,t)\in \{0,1\}\times (0,T); \quad y(x,0)=y_0(x),\,x\in (0,T), \] where \(T>0,\) \(\omega \subset\subset (0,1)\) is a non empty open interval, \(v\in L^\infty (\omega\times (0,T))\) is the control, \(y\) is the associated state and \(1_\omega\) is the characteristic function of the control set. The function \(D\) belongs to \(C^1([0,1])\) with \(D(x)\geq D_0>0\) and \(y_0\in L^2(0,1)\). The function \(f:\mathbb R\to\mathbb R\) is supposed to be at least locally Lipschitz-continuous.
The goal is achieved by means of standard fixed point formulation, gradient techniques and Newton-Raphson strategy. Some data for which least squares algorithm converge are exhibited.

MSC:

35Q93 PDEs in connection with control and optimization
49J05 Existence theories for free problems in one independent variable
65K10 Numerical optimization and variational techniques
93B05 Controllability
35B44 Blow-up in context of PDEs
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